From a group of 10 boys `B_1 , B_2 . . . B_10` and `G_1 , G_2,. . . ,G_5` the number of ways of selection of a group of 3 boys and 3 girls such that `B_1 and B_2` are not together in the group is ?

Answer

To solve the problem of selecting a group of 3 boys and 3 girls from a group of 10 boys (B1, B2, ..., B10) and 5 girls (G1, G2, ..., G5) such that B1 and B2 are not together, we can follow these steps:

### Step 1: Calculate the total number of ways to select 3 boys from 10 boys.
The total number of ways to select 3 boys from 10 can be calculated using the combination formula:
\[ \text{Total ways to select 3 boys} = \binom{10}{3} \]
Calculating this:
\[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

### Step 2: Calculate the number of ways to select 3 boys such that B1 and B2 are together.
If B1 and B2 are together, we can treat them as a single unit. Therefore, we have the following units to choose from:
- The unit (B1, B2)
- B3, B4, B5, B6, B7, B8, B9, B10 (8 boys left)

Now we need to select 1 more boy from these 8 remaining boys:
\[ \text{Ways to select 1 boy from 8} = \binom{8}{1} = 8 \]
Thus, the total number of ways to select 3 boys where B1 and B2 are together is:
\[ \text{Total ways with B1 and B2 together} = 8 \]

### Step 3: Calculate the number of ways to select 3 boys such that B1 and B2 are not together.
To find the number of ways to select 3 boys such that B1 and B2 are not together, we subtract the number of ways in which B1 and B2 are together from the total number of ways to select 3 boys:
\[ \text{Ways with B1 and B2 not together} = \binom{10}{3} - \text{Total ways with B1 and B2 together} = 120 - 8 = 112 \]

### Step 4: Calculate the number of ways to select 3 girls from 5 girls.
The number of ways to select 3 girls from 5 can be calculated similarly:
\[ \text{Total ways to select 3 girls} = \binom{5}{3} \]
Calculating this:
\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 5: Calculate the total number of ways to select the group.
Now, we can find the total number of ways to select 3 boys (where B1 and B2 are not together) and 3 girls:
\[ \text{Total ways} = (\text{Ways with B1 and B2 not together}) \times (\text{Ways to select 3 girls}) = 112 \times 10 = 1120 \]

### Final Answer
Thus, the total number of ways to select a group of 3 boys and 3 girls such that B1 and B2 are not together is:
\[ \boxed{1120} \]